Finite Group Databasebeta

← #6985965

Subgroups — #6985965

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#36535C211yesmaximal, characteristic, complemented
3#32823C212yesmaximal
4#32379C214yesmaximal, complemented
5#32231C214yesmaximal, complemented
6#30305C211yesmaximal, characteristic, complemented
7#30096C212yesmaximal, complemented
8#30066C212yesmaximal
9#29280C212yesmaximal, complemented
10#29208C211yesmaximal, characteristic, complemented
11#29121C211yesmaximal, characteristic
12#26091C212yesmaximal, complemented
13#26025C212yesmaximal, complemented
14#25059C212yesmaximal, complemented
15#24993C212yesmaximal, complemented
16#23815C211yesmaximal, characteristic
17#22018C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19#2868C2211yescharacteristic
20#2864C2212yes—
21#2859C2211yescharacteristic
22#2858C2212yes—
23#2590—22no—
24#2588—22no—
25#2581—21no—
26#2571C2211yescharacteristic
27#2570C2212yes—
28#2565—21no—
29#2552C2211yescharacteristic
30#2550C2212yes—
31#2547—21no—
32#2545—22no—
33#2538—21no—
34#2538C2212yes—
35#2512—22no—
36#2512C2214yes—
37#2370—21no—
38#2355—21no—
39#2355C2211yescharacteristic
40#2355C2211yescharacteristic
41#2354—21no—
42#2354C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×#127C2211yescharacteristic
46C4×#125C2212yes—
47C42×Q8C2211yescharacteristic
48#2101C2214yes—
49#2099C2211yescharacteristic
50#2099C2211yescharacteristic
51#2099C2212yes—
52#2092C2214yes—
53#2092C2214yes—
54#2085C2214yes—
55#2084C2214yes—
56#2073C2214yes—
57#2071C2212yes—
58#2067C2214yes—
59#2065C2214yes—
60#2058C2212yes—
61#2058C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2041C2211yescharacteristic
65#2041C2211yescharacteristic
66#2041C2212yes—
67#2025C2214yes—
68#2023C2212yes—
69#1990C2212yes—
70#1989C2214yes—
71#1984C2212yes—
72#1949C2214yes—
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1934C2214yes—
77#1926C2214yes—
78#1925C2214yes—
79#1921C2214yes—
80#1921C2214yes—
81#1920C2214yes—
82#1919C2214yes—
83#1916C2214yes—
84#1845C2212yes—
85#1844C2212yes—
86#1839C2212yes—
87#1834C2212yes—
88#1831C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94#1744C2211yescharacteristic
95#556—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#126—24no—
102C2×#125—21no—
103C2×#125C2312yes—
104C2×#124—22no—
105C2×#124—22no—
106C2×C22⋊Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#102C2312yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2312yes—
126C2×#102C2312yes—
127C2×#102C2312yes—
128C2×C4×C8C2312yes—
129C2×C4×C8C2312yes—
130#400—22no—
131#400—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#397—24no—
136#396—24no—
137#396—24no—
138#395—22no—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—22no—
143#391—22no—
144#390—21no—
145#390—21no—
146#390—22no—
147#390—22no—
148#390C2312yes—
149#390C2314yes—
150#390C2314yes—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389C2311yescharacteristic
159#389C2312yes—
160#389C2311yescharacteristic
161#389C2314yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—24no—
166#388—24no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#382—21no—
187#382—21no—
188#382C2312yes—
189#382C2312yes—
190#382C2312yes—
191#380—22no—
192#380—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C43C2312yes—
211C43C2311yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#336C2312yes—
223#336C2311yescharacteristic
224#336C2311yescharacteristic
225#336C2312yes—
226#336C2314yes—
227#336C2314yes—
228#336C2314yes—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8C2×D411yescharacteristic
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2412yes—
235C23×C4—22no—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2414yes—
239C4⋊Q8—22no—
240C4⋊Q8—41no—
241C4⋊Q8—44no—
242C4⋊Q8—44no—
243C4⋊Q8—82no—
244#127—42no—
245#126—416no—
246#125—24no—
247#125—42no—
248#125—44no—
249#124—48no—
250#124—84no—
251C22⋊Q8—48no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—48no—
256C4×Q8—44no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—44no—
260C4×D4—44no—
261#117—48no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—44no—
325#107—44no—
326#106—44no—
327#106—44no—
328C4⋊C8—24no—
329C4⋊C8—44no—
330#102—24no—
331#102—44no—
332#102—42no—
333#102—44no—
334#102—42no—
335#102—42no—
336C4×C8—24no—
337C4×C8—42no—
338#95—21no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—21no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359C24—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—81no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2511yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C42−1+412yes—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—24no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—41no—
470C42—42no—
471C42—42no—
472C42—41no—
473C42—48no—
474C42—42no—
475C42—42no—
476C42—82no—
477C42—82no—
478C42#13514yes—
479C42AGL(1,ℤ/8)14yes—
480C422−1+414yes—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C23×D411yescharacteristic
489Q8—44no—
490Q8—84no—
491Q8—88no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—84no—
497D4—84no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—88no—
545C2×C4#53612yes—
546C2×C4#53612yes—
547C2×C4C2×#13512yes—
548C2×C4C2×#13512yes—
549C2×C4C2×#13512yes—
550C2×C4C2×AGL(1,ℤ/8)12yes—
551C2×C4C2×AGL(1,ℤ/8)12yes—
552C2×C4C2×AGL(1,ℤ/8)12yes—
553C2×C4C2×2−1+411yescharacteristic
554C2×C4C2×2−1+412yes—
555C2×C4#58512yes—
556C2×C4#58511yescharacteristic
557C8—42no—
558C8—42no—
559C8—44no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22C2×#53611yescharacteristic
568C22#348711yescharacteristic
569C22#348911yescharacteristic
570C22C22×#13511yescharacteristic
571C22C22×AGL(1,ℤ/8)11yescharacteristic
572C22#358511yescharacteristic
573C22#358611yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—81no—
586C4—82no—
587C4—82no—
588C4—81no—
589C4—81no—
590C4—84no—
591C4#322814yes—
592C4#355512yes—
593C4#356512yes—
594C2—162no—
595C2#4289111yescharacteristic
596C2C2×#322811yescharacteristic
597C2#6088611yescharacteristic
598C2#6188211yescharacteristic
599C2#6189011yescharacteristic
600C2#6210211yescharacteristic
601C2#6210711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #36535C2semidirect
4. #32379C2semidirect
5. #32231C2semidirect
6. #30305C2semidirect
7. #30096C2semidirect
9. #29280C2semidirect
10. #29208C2semidirect
12. #26091C2semidirect
13. #26025C2semidirect
14. #25059C2semidirect
15. #24993C2semidirect
17. #22018C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365942
#365352
#328232
#323792
#322312
#303052
#300962
#300662
#292802
#292082
#291212
#260912
#260252
#250592
#249932
#238152
#220182